Related papers: Horocyclic Brunn-Minkowski inequality
We prove a Minkowski type inequality for weakly mean convex and star-shaped hypersurfaces in warped cylinders which are asymptotically flat or hyperbolic. In particular, we show that this sharp inequality holds for outward minimizing…
Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total…
For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and…
We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the…
This article derives closed-form parametric formulas for the Minkowski sums of convex bodies in d-dimensional Euclidean space with boundaries that are smooth and have all positive sectional curvatures at every point. Under these conditions,…
The Brunn-Minkowski theory in convex geometry concerns, among other things, the volumes, mixed volumes, and surface area measures of convex bodies. We study generalizations of these concepts to Borel measures with density in…
In this paper we investigate the existence and uniqueness of spacelike radial graphs of prescribed mean curvature in the Lorentz-Minkowski space $\mathbb{L}^{n+1}$, for $n\geq 2$, spanning a given boundary datum lying on the hyperbolic…
We quantify Prokhorov's Theorem by establishing an explicit formula for the Hausdorff measure of non-compactness (HMNC) for the parametrized Prokhorov metric on the set of Borel probability measures on a Polish space. Furthermore, we…
We follow the method of ABP estimate in \cite{brendle2021} and apply it to spacelike submanifolds in $\mathbb R^{n,1}$. We then obtain Michael-Simon type inequalities. Surprisingly, our investigation leads to a Sobolev inequality without a…
Time functions with asymptotically hyperbolic geometry play an increasingly important role in many areas of relativity, from computing black-hole perturbations to analyzing wave equations. Despite their significance, many of their…
In \cite{LX}, the first author and the third author introduced and studied the horospherical $p$-Minkowski problem for smooth horospherically convex domains in hyperbolic space. In this paper, we introduce and solve the discrete…
In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space $\mathbb{H}^{n+1}$ based on the locally constrained inverse curvature flow introduced…
In 2011 Lutwak, Yang and Zhang extended the definition of the $L_p$-Minkowski convex combination ($p \geq 1$) introduced by Firey in the 1960s from convex bodies containing the origin in their interiors to all measurable subsets in…
For a regular surface in Euclidean space $\mathbb{R}^3$, umbilic points are precisely the points where the Gauss and mean curvatures $K$ and $H$ satisfy $H^2=K$; moreover, it is well-known that the only totally umbilic surfaces in…
In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs…
In this note we derive a new Minkowski-type inequality for closed convex surfaces in the hyperbolic 3-space. The inequality is obtained by explicitly computing the area of the family of surfaces obtained from the normal flow and then…
General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures…
In this paper, we can obtain curvature estimates for spacelike admissible graphic hypersurfaces in the $(n+1)$-dimensional Lorentz-Minkowski space $\mathbb{R}^{n+1}_{1}$, and through which the existence of spacelike admissible graphic…
In this paper, we study on three kinds of spacelike helicoidal surfaces in Minkowski $4$--space. First, we give an isometry between such helicoidal surfaces and rotational surfaces which is a kind of generalization of Bour theorem in…
According to the holographic principle, the description of a volume of space can be thought of as encoded on its boundary. Holographic principle establishes equivalence, or duality, between theoretical description of volume physics, which…